미분방정식 전체 문제

We want to know when

$$\begin{aligned} [T(0)-A]e^{kt}+A&=A \\ \text{That is, when} \\ [T(0)-A]e^{kt}&=0 \\ \text{Since $e^{kt}>0$ for all values of $k$ and $t$, this happens exactly when} \\ T(0)-A=0 \end{aligned}$$

So: if the initial temperature of the object is not the same as the ambient temperature, then according to this model, it never will be! (However, as $t$ gets larger and larger, $T(t)$ gets closer and closer to $A$–it just never exactly reaches there.)

If the initial temperature of the object starts out the same as the ambient temperature, then $T(t)=A$ for all values of $t$.

Since $b$ is a positive constant, $\displaystyle \lim_{t \to \infty}e^{bt}=\infty$. Therefore:

$$\begin{aligned} \lim_{t \to \infty}P(t)&=\lim_{t \to \infty} P(0)e^{bt} =\left\{\begin{array}{cc} 0&\text{ if }P(0)=0\\ \infty&\text{ if }P(0)>0 \end{array}\right. \end{aligned}$$

If $P(0)=0$, then the model simply says “a population that starts with no individuals continues to have no individuals indefinitely," which certainly makes sense. If $P(0) \neq 0$, then (since we can't have a negative population) $P(0)>0$, and the model says “a population that starts out with some individuals will end up with any gigantically huge number you can think of, given enough time." This one doesn't make so much sense. Populations only grow to a certain finite amount, due to scarcity of resources and such. In the derivation of the Malthusian model, we assume a constant net birth rate–that the birth and death rates (per individual) don't depend on the population, which is not a reasonable assumption long-term.

Let $Q(t)$ be the quantity of morphine in a patient's bloodstream at time $t$, where $t$ is measured in minutes.

Using the definition of a derivative, $$\frac{\mathrm{d}Q}{\mathrm{d}t}=\lim_{h \to 0}\frac{Q(t+h)-Q(t)}{h}\approx \frac{Q(t+1)-Q(t)}{1}$$ So, $\frac{\mathrm{d}Q}{\mathrm{d}t}$ is roughly the change in the amount of morphine in one minute, from $t$ to $t+1$.

The sentence tells us that the change in the amount of morphine in one minute is about $-0.003Q$, where $Q$ is the quantity in the bloodstream. That is: $$\frac{\mathrm{d}Q}{\mathrm{d}t}=-0.003Q(t)$$

If $p(t)$ is the proportion of times speakers use the new form, measured between 0 and 1, then $1-p(t)$ is the proportion of times speakers use the old form.

The law, then, states that $\frac{\mathrm{d}p}{\mathrm{d}t}$ is proportional to $p(t)\times \big(1-p(t)\big)$. When we say two quantities are proportional, we mean that one is a constant multiple of the other. So, the law says $$\frac{\mathrm{d}p}{\mathrm{d}t}=\alpha p(t)\big(1-p(t)\big)$$ for some constant $\alpha$.

Remark: it follows from this model that, when a new form is either very rare or entirely ubiquitous, the rate of change of its adoption is small. This makes sense: if the new form is used all the time ($p(t)\approx 1$), there's nobody left to convert; if the new form is almost never used ($p(t)\approx 0$) then people don't know about it, so they won't pick it up.